How to find Steiner minimal trees in euclideand-space

Abstract

Abstract. This paper has two purposes. The first is tO present a new way to find a Steiner minimum tree (SMT) connecting N sites in d-space, d > 2. We present (in Appendix 1) a computer code for this purpose. This is the only procedure known to the author for finding Steiner minimal trees in d-space for d > 2, and also the first one which fits naturally into the framework of "backtracking" and "branch-and-bound." Finding SMTs of up to N = 12 general sites in d-space (for any d) now appears feasible.We tabulate Steiner minimal trees for many point sets, including the vertices of most of the regular and Archimedean d-polytopes with _< 16 vertices. As a consequence of these tables, the Gilbert-Pollak conjecture is shown to be false in dimensions 3 9. (The conjecture remains open in other dimensions; it is probably false in all dimensions d with d > 3, but it is probably true when d = 2.)The second purpose is to present some new theoretical results regarding the asymptotic computational complexity of finding SMTs to precision e.We show that in two-dimensions, Steiner minimum trees may be found exactly in exponential time One therefore suspects that this problem may be solved exactly in polynomial time. We show that this suspicion is in fact true--if a certain conjecture about the size of "Steiner sensitivity diagrams" is correct. All of these algorithms are for a "real RAM" model of computation allowing infinite precision arithmetic. They make no probabilistic or other assumptions about the input; the time bounds are valid in the worst case; and all our algorithms may be implemented with a polynomial amount of space. Only algorithms yielding the exact optimum SMT, or trees with lengths < (1 + e) x optimum, where e is arbitrarily small, are considered here. Key Words. Steiner trees, Gilbert-Pollak conjecture, Subexponential algorithms, Regular polytopes, Sensitivity diagrams. Basic Facts and Definitions.This section presents basic facts and definitions which we will need. Readers familiar with Steiner tree terminology may skip to Section 1.A set of real numbers is said to be in "general position" if they satisfy no nontrivial algebraic equation with integer coefficients. A positive real-valued function f(N), which may be bounded for all sufficiently large N by Received November, 1988; revised September, 1989, and February, 1990. Communicated by F. K. Hwang. 138 W.D. Smith where 1 < C a < C 2, is said to be "exponential." A function which grows faster than any upper bound of this type is "superexponential," and one which cannot be bounded by any lower bound of this type is "subexponential." Z and R represent the sets of integers and real numbers, respectively.The Steiner minimal tree (SMT) on N particles 21..-s in d-space, d > 2, is the shortest tree containing the sites. A "Steiner tree" is a tree of line segments with the following properties:1. It contains N "regular sites" 21'")~u, and possibly K additional "Steiner nodes" ~N+ 1 "'" 2N+K" 2. Each Steiner node has valence 3, and the edges emanating&ore it lie in a p...

Steiner Tree Heuristic in the Euclidean d-Space Using Bottleneck Distances

“…6. The green plot for n = 10 is the average ratio and CPU time of the exact solutions [14]. Once again, the DB-heuristic outperforms the DMheuristic when comparing the quality of solutions.…”

Monte Carlo Tree Search Algorithm for the Euclidean Steiner Tree Problem

“…Another good review of Steiner trees applications can be found in [5]. Although exact algorithms exist that enable to solve ESTP on a plane, such as those proposed by Melzak [9], [10], Trietsch and Hwang [11] and Smith [12], they are time-consuming, and there is a need for new algorithms. New heuristics are still being proposed [13].…”

Approximations and Lower Bounds for the Length of Minimal Euclidean Steiner Trees

“…However, finding lower bounds is not so easy. In this paper we generalise an observation of Smith [7] for four terminals, to give a general lower bound for arbitrary sets of terminals, in terms of toroidal images. We then use this to give more specific lower bounds based on angles created by the edges of an approximating tree.…”

“…see Chang [4] and Beasley [2]. In particular, Smith [7] gave an approximation algorithm for minimum Steiner trees in d-dimensional Euclidean space which is practical for small sets of terminals and small d. However, no guaranteed performance ratio at all has been published for such algorithms.…”

“…Specifically, in d-dimensional Euclidean space the following criterion was used in [7] for when a full tree T is a close enough approximation to a Steiner tree with the same topology: for some predetermined > 0, all angles formed by the edges at the Steiner points of T are at least 120…”